Question: $ -2.\overline{7} \div 0.\overline{2} = {?} $
Answer: First convert the repeating decimals to fractions. $\begin{align*} 10x &= -27.7778...\\ x &= -2.7778...\end{align*} $ $\begin{align*} 9x &= -25 \\ x &= -\dfrac{25}{9}\end{align*} $ $\begin{align*} 10y &= 2.2222...\\ y &= 0.2222...\end{align*} $ $\begin{align*} 9y &= 2 \\ y &= \dfrac{2}{9}\end{align*} $ So, the problem becomes: $ -\dfrac{25}{9} \div \dfrac{2}{9} = {?} $ Dividing by a fraction is the same as multiply by the reciprocal of that fraction. $ -\dfrac{25}{9} \times \dfrac{9}{2} = {?} $ $ \phantom{-\dfrac{25}{9} \times \dfrac{2}{9}} = \dfrac{-25 \times 9}{9 \times 2} $ $ \phantom{-\dfrac{25}{9} \times \dfrac{2}{9}} = \dfrac{-25 \times \cancel{9}} {\cancel{9} \times 2} $ $ \phantom{-\dfrac{25}{9} \times \dfrac{2}{9}} = -\dfrac{25}{2} $